Systolic lattice extensions of classical Schottky groups

Junzhi Huang; Matthew Zevenbergen

REVIEW 2 major objections 3 minor 3 references

Systolic lattice extensions of classical Schottky groups

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that every classical Schottky subgroup of the isometry group of hyperbolic $d$-space admits a torsion-free lattice extension whose short loxodromic elements are all conjugate into the subgroup.

desk verdict Substantial new extension of Agol's inbreeding to all classical Schottky subgroups, with a plausible proof and two fixable gaps (over-k perturbability and a d=3 commensurability overreach). read the letter →

arxiv 2505.24118 v2 pith:NB3QZURU submitted 2025-05-30 math.GT

Junzhi Huang , Matthew Zevenbergen This is my paper

classification math.GT MSC 20H1022E4030F4051M1057M50

keywords classicalSchottkygroupssystoliclatticeextensionshyperbolicmanifoldscomplextranslationlengthholonomySalemnumbersarithmeticlatticesChabautytopology

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Classical Schottky groups are free groups generated by isometries that pair disjoint half-spaces in hyperbolic space. The paper proves that, for any admissible pair $(k,q)$ (a totally real number field and a quadratic form of signature $(d,1)$ whose other conjugates are positive definite) and any $D>0$, every classical Schottky subgroup $F\leq \mathrm{SO}'(q,k)$ is contained in a torsion-free lattice $\Gamma$ with a systolic property: every loxodromic element of translation length at most $D$ in $\Gamma$ is conjugate into $F$. When $q$ is anisotropic the lattice can be taken cocompact. These extensions make every short closed geodesic of the quotient manifold descend from the Schottky quotient, which the authors exploit to show that complex systoles of closed hyperbolic manifolds are dense and to produce non-arithmetic lattices with prescribed Salem-number systoles.

What carries the argument

The machinery is a cut-and-paste ('inbreeding') construction on an arithmetic manifold, adapted from Agol's construction of short-systole hyperbolic $4$-manifolds. Starting from $\Gamma_1=\mathrm{SO}'(q,\mathcal{O}_k)$, the proof passes to finite-index subgroups $\Gamma_2$, $\Gamma_3$ that are torsion-free, have no loxodromic elements of length $\leq D$, and contain embedded totally geodesic hypersurfaces $\Sigma_i$ with stabilizers $K_i$; it then cuts the quotient along the $\Sigma_i$ and glues the inner boundary components by the Schottky generators $g_i$ through the Klein–Maskit combination theorem, before gluing the outer boundary components to obtain a finite-volume manifold. The systolic estimate is carried by two compact sets: $C_1$, a fundamental domain for the $F$-action on an $R$-neighborhood of the convex hull of the limit set, and $C_2$, a $D$-neighborhood of the truncated hypersurface pieces; separability of geometrically finite subgroups ensures both embed in the intermediate covers. Any geodesic of length $\leq D$ in the final manifold must cross the glued hypersurfaces, and the choice of $C_1,C_2$ forces its lift into $C_1$, so it projects through $F\setminus\mathbb{H}^d$ and is conjugate into $F$.

What would settle it

Take $d=3$, $k=\mathbb{Q}(\sqrt{2})$, $q(x)=x_0^2+x_1^2+x_2^2-\sqrt{2}\,x_3^2$, choose a classical Schottky generator $g$ with $\ell(g)<\varepsilon$ as in Corollary 3, run the construction, and search the quotient for a closed geodesic of length $<\varepsilon$ whose lift avoids the embedded copy of $C_1$; finding one would refute the $D$-systolic conclusion of the Main Theorem.

Watch

Extended reading notes

Core claim

The central claim is the Main Theorem: if $(k,q)$ is admissible and $D>0$, every classical Schottky subgroup $F\leq \mathrm{SO}'(q,k)$ admits a torsion-free $D$-systolic lattice extension $\Gamma\leq \mathrm{SO}'(q,k)$, and if $q$ is anisotropic the extension can be chosen cocompact. A $D$-systolic extension is one in which every loxodromic $g\in\Gamma$ with translation length $\ell(g)\leq D$ is conjugate in $\Gamma$ into $F$, so all closed geodesics of length at most $D$ in $\Gamma\setminus\mathbb{H}^d$ descend from closed geodesics in $F\setminus\mathbb{H}^d$. The proof builds $\Gamma$ by cutting a finite-index arithmetic manifold along embedded totally geodesic hypersurfaces and regluing the pieces with the Schottky generators using Klein–Maskit combination, then closing the remaining boundary components; the systolic conclusion follows because any short geodesic must lift to a compact set $C_1$ whose projection factors through $F\setminus\mathbb{H}^d$. Consequences include density of complex systoles in $\mathbb{R}_+\times C(\mathrm{SO}(d-1))$ for $d\geq 3$, the construction of hyperbolic $d$-manifolds with systole $\log(\lambda)$ for any Salem number $\lambda$, and a new route to non-arithmetic lattices.

Load-bearing premise

The load-bearing assumption is that the half-spaces in a standard generating set of a classical Schottky group can be perturbed so every boundary hyperplane is defined over the number field $k$; the paper asserts this without proof or reference, and the rest of the argument depends on it to make each hyperplane stabilizer an arithmetic lattice.

Editorial extensions

If this is right

Applied to a cyclic group $\langle g\rangle$, the Main Theorem turns any loxodromic $g\in\mathrm{SO}'(q,k)$ into the systole of a cocompact lattice when $q$ is anisotropic, which is the engine for the density results.
For every $d\geq 3$, the set of complex systoles of closed hyperbolic $d$-manifolds is dense in $\mathbb{R}_+\times C(\mathrm{SO}(d-1))$, so arbitrarily short geodesics with arbitrary holonomy, including infinite-order holonomy, occur.
Every classical Schottky subgroup is a Chabauty limit of cocompact lattices, and for $d\geq 3$ the space of discrete torsion-free subgroups is not locally connected at such a subgroup.
For any Salem number $\lambda$ and any $d\geq 3$, there is a non-arithmetic, quasi-arithmetic hyperbolic $d$-manifold whose systole is exactly $\log(\lambda)$.
Proposition 5 shows that in arithmetic lattices of simplest type there are only finitely many holonomies for loxodromics of a fixed Salem length $\log(\lambda)$; under the Boyd–Salem conjecture, their complex translation lengths form a discrete set, in contrast with the dense set for general lattices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The unproved perturbability step (assuming half-spaces can be chosen with normals in $k^{d+1}$) is the main point to stress-test: if it fails for a particular standard generating set, the theorem may still hold for a conjugate or a different generating set, but the present proof would not cover that case.
Remark 1 suggests a testable refinement: for any $n$, the same construction should yield closed hyperbolic manifolds whose complex systoles are dense and with exactly $n$ systole-realizing geodesics; verifying this would strengthen Corollary 4.
Corollary 6 avoids the usual short-geodesic criterion for non-arithmeticity, so an independent check via invariant trace fields or commensurators would both confirm the method and potentially answer the authors' Question 1 about incommensurability with inbred manifolds.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear [Colleague],

Main take: this is a substantial new result. Huang and Zevenbergen extend Agol's inbreeding construction to produce D-systolic lattice extensions for every classical Schottky subgroup of SO'(q,k), with cocompactness when q is anisotropic. That is a real advance: prior work covered the whole lattice or specified cyclic groups, not arbitrary Schottky subgroups. The corollaries are also new: density of complex systoles for closed hyperbolic d-manifolds (d≥3), first examples of arbitrarily short geodesics with infinite-order holonomy for d≥4, and a construction of non-arithmetic lattices with prescribed Salem systole.

The proof is dense but clearly structured, and the reliance on standard tools (Borel–Harish-Chandra, Selberg, separability, Klein–Maskit) is legitimate. The main theorem seems likely correct. The self-citation pattern is fine: [Zev24] is used for the non-local-connectedness claim, and [DH24] is the result being extended; the Main Theorem is logically independent.

Two soft spots. First, the perturbability assertion: after fixing a standard generating set, they say 'By perturbing each A_i we can assume all half-spaces are defined over k.' This is load-bearing, because it lets them apply Borel–Harish-Chandra to the hyperplane stabilizers. No argument is given. The claim is true: the space of valid halfspace tuples for the fixed generators is open, and k-defined hyperplanes are dense, so a perturbation exists. But the written proof should supply this; as it stands, it's a gap in exposition, not a serious flaw. Second, the proof of Corollary 6 uses Gromov–Piatetski-Shapiro to assert that an arithmetic lattice must be commensurable to SO'(q,O_k), hence of simplest type. For d=3 this is overstrong: there are arithmetic hyperbolic 3-manifolds of non-simplest type (e.g., from quaternion algebras over imaginary quadratic fields). The argument likely works for d≥4, but the corollary states d≥3. The authors need to either restrict the statement or add an argument ruling out non-simplest-type arithmeticity for d=3.

The systolic argument itself is compressed but plausible; an expert referee should verify the geometric claims about short geodesics having to lie in the neighborhood of the convex core, but they seem to follow from the stated separation properties.

Who this is for: people working on hyperbolic manifolds, systolic geometry, and arithmetic lattices. It deserves serious refereeing. I'd recommend accepting the paper pending the two clarifications.

Referee Report

2 major / 3 minor

Summary. The paper introduces the notion of a D-systolic lattice extension: a lattice Γ containing a discrete group G such that every loxodromic element of Γ with translation length at most D is conjugate into G. The Main Theorem asserts that for any admissible pair (k,q) and any D>0, every classical Schottky subgroup F of SO'(q,k) admits a torsion-free D-systolic lattice extension, cocompact when q is anisotropic. The proof starts from the arithmetic lattice SO'(q,O_k), uses residual finiteness, separability, Selberg's lemma, and Klein-Maskit combination to construct an extension, and then uses a cut-and-paste argument on the associated hyperbolic manifold to obtain the systolic property. The paper also derives corollaries: density of Schottky groups admitting cocompact lattice extensions, Chabauty approximation of Schottky groups by cocompact lattices, density of complex systoles of closed hyperbolic d-manifolds for d≥3, and construction of non-arithmetic lattices with prescribed Salem-number systole.

Significance. If the main proof is completed, the Main Theorem is a substantial result: it extends Brooks-type lattice-approximation to a dense family of Schottky groups in all dimensions, and it gives a strong systolic control that goes beyond previous Agol-type constructions. The corollaries on complex systoles and on non-arithmetic lattices are attractive and well-motivated. The paper is written with standard tools and is largely self-contained, but one load-bearing step in the proof of the Main Theorem is currently asserted without proof. The overall significance is high, conditional on repairing that step.

major comments (2)

[Section 3, proof of Main Theorem, first paragraph] The sentence "By perturbing each A_i we can assume that all the half spaces are defined over k, i.e. P_i = (v_i)^\perp for some v_i in k^{d+1}" is unproved and load-bearing. It is used immediately to conclude, via the Borel-Harish-Chandra theorem, that stab_{Γ1}(P_i) is a lattice in Isom(P_i), and this is the basis for the finite-index subgroup construction in Claim 1. The assertion is not automatic for an arbitrary standard generating set of a Schottky group F ≤ SO'(q,k): only the generators are known to lie in SO'(q,k), while the supplied half-spaces are only known to be defined over R. Moreover, the identities g_i(A_{-i}) = H^d \ A_i must be preserved for the fixed generators, so the perturbation must move the pairs (A_{-i}, A_i) coherently. The missing argument is an openness/density statement in the space of Schottky half-space configurations with fixed generators, together with density of k-defined hyperplanes in that space. This should be stated as a lemma and proved, or replaced by a different argument, before the Borel-Harish-Chandra step is valid.
[Section 4.1, proof of Corollary 2] The final statement that D_d is not locally connected at F relies on the assertion that for d≥3, the conjugacy classes of cocompact lattices are connected components of D_d, cited to [Zev24, Thm. A]. This is an arXiv preprint by the second author, and it is load-bearing for the strongest part of Corollary 2. The manuscript should either state explicitly that this corollary is conditional on [Zev24], or provide a proof of the needed connected-components statement. As written, the reader cannot verify the non-local-connectedness conclusion from the present paper alone.

minor comments (3)

[Section 3, Claim 1] The proof of property (A6) is compressed: the "ping-pong argument" is mentioned without details, although (A6) is used later in the final systole argument. Since the intended argument is plausible, a short expanded explanation or a reference to a precise lemma would improve the paper.
[Section 3, final paragraph] The sentence "Γ is cocompact if and only if Γ1 is" is asserted without justification. The forward implication (cocompact Γ1 implies cocompact Γ) is clear from the construction, but the converse is not used for the theorem and should be omitted or given a one-sentence explanation.
[Section 4.1, proof of Corollary 2] In the last line of the proof, "Fn → F in Dn" should presumably read "in D_d".

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the Main Theorem is derived from external theorems, with only a minor non-load-bearing self-citation and one genuine proof gap that is not circular.

full rationale

The derivation chain for the Main Theorem is self-contained relative to the literature it invokes: Borel–Harish-Chandra gives Γ1=SO'(q,Ok) as a lattice; residual finiteness and Selberg's lemma provide a torsion-free finite-index subgroup with systole greater than D; Bergeron–Haglund–Wise separability plus Scott's criterion yield the subgroup Γ3 containing K with C2 embedded; and Klein–Maskit combination theorems produce the final discrete extension Γ6. None of these steps re-uses the theorem being proved or fits a parameter to the target. The only overlapping-author citations are [DH24], the density-of-systoles result being extended and used only as motivation, and [Zev24, Thm. A], which supports the connected-component fact in the proof of Corollary 2; neither is needed to establish the Main Theorem, so the self-citation is not load-bearing. I note one genuine non-circular gap: the proof asserts, 'By perturbing each A_i we can assume that all the half spaces are defined over k, i.e. P_i = (v_i)^\perp for some v_i in k^{d+1}' (Section 3), and immediately uses this to apply Borel–Harish-Chandra to stabilizers of P_i. No openness/density argument is supplied for this perturbability, so the written proof has a gap; however, this is a missing justification, not a circular reduction, because it neither defines the conclusion in terms of the hypothesis nor fits a parameter to the target. Score 1 reflects the minor, non-load-bearing self-citation rather than any circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claim rests on standard deep results in arithmetic groups and hyperbolic manifold theory plus one unproved perturbation assumption about half-spaces over k. There are no fitted free parameters and no invented geometric or algebraic entities.

assumptions (9)

standard math Borel-Harish-Chandra theorem: for admissible (k,q), SO'(q,O_k) is a lattice in SO'(q,R), cocompact when q is anisotropic.
Invoked in Sections 2.2 and 3 to establish that Gamma_1 and hyperplane stabilizers are lattices.
standard math Selberg's lemma: finitely generated linear groups have torsion-free finite-index subgroups.
Used in Claim 1 to arrange property (A1) that Gamma_2 is torsion-free.
standard math Residual finiteness of arithmetic lattices and separability of cyclic subgroups.
Used in Claim 1 to arrange (A2) and (A3), eliminating short loxodromic elements and embedding a compact core.
standard math Quasiconvex subgroup separability in arithmetic hyperbolic lattices (Bergeron-Haglund-Wise; Wise; Bergeron-Wise).
Used in Claim 2 to find Gamma_3 with the desired embedding properties for C_2 and the hypersurfaces.
standard math Klein-Maskit combination theorems.
Used to glue the cut manifolds and to prove that Gamma_5 is discrete and torsion-free.
standard math Scott's separability criterion for subgroups.
Converts separability of K into the existence of a finite-index subgroup Gamma_3 realizing the embedding of C_2.
standard math Emery-Ratcliffe-Tschantz results on Salem numbers and arithmetic hyperbolic groups (their Thm. 5.2 and Lemmas 4.2, 4.5).
Used in Proposition 5 to restrict the field of definition and the possible characteristic polynomials of loxodromic elements.
standard math Futer-Purcell-Schleimer Proposition 1.4 on Margulis tube distance.
Used to choose epsilon so that short geodesics avoid epsilon-thin cusp neighborhoods in the systolic argument.
ad hoc to paper The half-spaces defining a classical Schottky group can be perturbed to be defined over k while keeping the same generators.
Asserted without proof in Section 3; it is load-bearing because it makes hyperplane stabilizers arithmetic lattices defined over k.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Systolic lattice extensions of classical Schottky groups." pith.science (2026). https://pith.science/paper/NB3QZURU

@misc{pith2026250524118,
  author       = {Pith},
  title        = {Pith review of: Systolic lattice extensions of classical Schottky groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NB3QZURU}},
  note         = {Machine review of arXiv:2505.24118}
}

abstract

We produce lattice extensions of a dense family of classical Schottky subgroups of the isometry group of $d$-dimensional hyperbolic space. The extensions produced are said to be systolic, since all loxodromic elements with short translation length are conjugate into the Schottky groups. Various corollaries are obtained, in particular showing that for all $d\geq3$, the set of complex translation lengths realized by systoles of closed hyperbolic $d$-manifolds is dense inside the set of all possible complex translation lengths. We also consider complex translation lengths in arithmetic hyperbolic $d$-manifolds, and provide a new way to construct non-arithmetic lattices.

Figures

Figures reproduced from arXiv: 2505.24118 by the authors.

**Figure 1.** The compact set C1 and its images are shown in blue, and the compact set C2 and its images are shown in green. Note that the image of C1 in F\Hd is NR [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[535]

Wise,Hyperplane sections in arithmetic hyperbolic manifolds, J

MR147566 [BHW11] Nicolas Bergeron, Fr´ ed´ eric Haglund, and Daniel T. Wise,Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc. (2) 83 (2011), no. 2, 431–448. MR2776645 [Bow09] Lewis Bowen, Free groups in lattices, Geom. Topol. 13 (2009), no. 5, 3021–3054. MR2546620 [Bow93] B. H. Bowditch, Geometrical finiteness for hyperbolic grou...

work page 2011
[565]

Serre, A course in arithmetic , Graduate Texts in Mathematics, vol

MR494062 [Ser73] J.-P. Serre, A course in arithmetic , Graduate Texts in Mathematics, vol. No. 7, Springer-Verlag, New York- Heidelberg, 1973. Translated from the French. MR344216 [Tho16] Scott Thomson, Quasi-arithmeticity of lattices in PO(n, 1), Geom. Dedicata 180 (2016), 85–94. MR3451458 [Thu78] William Thurston, The geometry and topology of 3-manifold...

work page arXiv 2016
[1992]

MR1219310 [Bro86] Robert Brooks, Circle packings and co-compact extensions of Kleinian groups , Invent. Math. 86 (1986), no. 3, 461–469. MR860677 [BT11] Mikhail V. Belolipetsky and Scott A. Thomson, Systoles of hyperbolic manifolds, Algebr. Geom. Topol. 11 (2011), no. 3, 1455–1469. MR2821431 [BW12] Nicolas Bergeron and Daniel T. Wise, A boundary criterion...

work page arXiv 1986

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [535]

Wise,Hyperplane sections in arithmetic hyperbolic manifolds, J

MR147566 [BHW11] Nicolas Bergeron, Fr´ ed´ eric Haglund, and Daniel T. Wise,Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc. (2) 83 (2011), no. 2, 431–448. MR2776645 [Bow09] Lewis Bowen, Free groups in lattices, Geom. Topol. 13 (2009), no. 5, 3021–3054. MR2546620 [Bow93] B. H. Bowditch, Geometrical finiteness for hyperbolic grou...

work page 2011

[2] [565]

Serre, A course in arithmetic , Graduate Texts in Mathematics, vol

MR494062 [Ser73] J.-P. Serre, A course in arithmetic , Graduate Texts in Mathematics, vol. No. 7, Springer-Verlag, New York- Heidelberg, 1973. Translated from the French. MR344216 [Tho16] Scott Thomson, Quasi-arithmeticity of lattices in PO(n, 1), Geom. Dedicata 180 (2016), 85–94. MR3451458 [Thu78] William Thurston, The geometry and topology of 3-manifold...

work page arXiv 2016

[3] [1992]

MR1219310 [Bro86] Robert Brooks, Circle packings and co-compact extensions of Kleinian groups , Invent. Math. 86 (1986), no. 3, 461–469. MR860677 [BT11] Mikhail V. Belolipetsky and Scott A. Thomson, Systoles of hyperbolic manifolds, Algebr. Geom. Topol. 11 (2011), no. 3, 1455–1469. MR2821431 [BW12] Nicolas Bergeron and Daniel T. Wise, A boundary criterion...

work page arXiv 1986